Multi-type Display Calculus for Semi-De Morgan Logic Fei Liang 1 , 2 - - PowerPoint PPT Presentation

Multi-type Display Calculus for Semi-De Morgan Logic

Fei Liang1,2 joint work with: G. Greco1, M. A. Moshier3 and A. Palmigiano1,4

1Delft University of Technology, the Netherlands 2Sun Yat-sen University, China 3Chapman University, California, USA 4University of Johannesburg, South Africa

TACL, Prague, 29th June, 2017

Motivation and Aim

◮ Sankappanavar H P

. Semi-De Morgan algebras[J]. The Journal of symbolic logic, 1987, 52(3): 712-724

◮ a common abstraction of De Morgan algebras and distributive

pseudo-complemented lattices

Motivation and Aim

◮ Sankappanavar H P

. Semi-De Morgan algebras[J]. The Journal of symbolic logic, 1987, 52(3): 712-724

◮ a common abstraction of De Morgan algebras and distributive

pseudo-complemented lattices

• Preminimal ¬(a ∨b) = ¬a ∧¬b,¬0 = 1
• Quasi-Minimal a ≤ ¬¬a
• Minimal a ∧b ≤ c ⇒ a ∧¬c ≤ ¬b
• Heyting a ∧¬a ≤ 0
• semi-De Morgan

¬a = ¬¬¬a,¬1 = 0 ¬¬a ∧¬¬b = ¬¬(a ∧b)

• a ≤ ¬¬a quasi-De Morgan
• ¬¬a ≤ a De Morgan
• Boolean

Motivation and Aim

◮ Sankappanavar H P

. Semi-De Morgan algebras[J]. The Journal of symbolic logic, 1987, 52(3): 712-724

◮ a common abstraction of De Morgan algebras and distributive

pseudo-complemented lattices

• Preminimal ¬(a ∨b) = ¬a ∧¬b,¬0 = 1
• Quasi-Minimal a ≤ ¬¬a
• Minimal a ∧b ≤ c ⇒ a ∧¬c ≤ ¬b
• Heyting a ∧¬a ≤ 0
• semi-De Morgan

¬a = ¬¬¬a,¬1 = 0 ¬¬a ∧¬¬b = ¬¬(a ∧b)

• a ≤ ¬¬a quasi-De Morgan
• ¬¬a ≤ a De Morgan
• Boolean

◮ Ma M, Liang F. Sequent Calculi for Semi-De Morgan and De Morgan

Algebras[J]. arXiv preprint:1611.05231, 2016.

Motivation and Aim

Is there an uniform way to deal with semi De Morgan negation and preserve real subformula property?

Motivation and Aim

Is there an uniform way to deal with semi De Morgan negation and preserve real subformula property?

◮ The answer is “Yes”, via multi-type methodology!

Preliminaries

De Morgan and semi-De Morgan Algebras

Definition

If (A,∨,∧,⊤,⊥) is a bounded distributive lattice, then an algebra A = (A,∨,∧,¬,⊤,⊥) is: for all a,b ∈ A, De Morgan algebra semi-De Morgan algebra ¬(a ∨b) = ¬a ∧¬b ¬(a ∨b) = ¬a ∧¬b ¬(a ∧b) = ¬a ∨¬b ¬¬(a ∧b) = ¬¬a ∧¬¬b ¬¬a = a ¬¬¬a = ¬a ¬⊥ = ⊤,¬⊤ = ⊥ ¬⊥ = ⊤, ¬⊤ = ⊥

De Morgan and semi-De Morgan Algebras

Definition

If (A,∨,∧,⊤,⊥) is a bounded distributive lattice, then an algebra A = (A,∨,∧,¬,⊤,⊥) is: for all a,b ∈ A, De Morgan algebra semi-De Morgan algebra ¬(a ∨b) = ¬a ∧¬b ¬(a ∨b) = ¬a ∧¬b ¬(a ∧b) = ¬a ∨¬b ¬¬(a ∧b) = ¬¬a ∧¬¬b ¬¬a = a ¬¬¬a = ¬a ¬⊥ = ⊤,¬⊤ = ⊥ ¬⊥ = ⊤, ¬⊤ = ⊥

Fact

A semi-De Morgan algebra A is a De Morgan algebra if and only if A satisfies the equation a ∨b = ¬(¬a ∧¬b) = ¬¬(a ∨b).

De Morgan and semi-De Morgan Algebras

Definition

If (A,∨,∧,⊤,⊥) is a bounded distributive lattice, then an algebra A = (A,∨,∧,¬,⊤,⊥) is: for all a,b ∈ A, De Morgan algebra semi-De Morgan algebra ¬(a ∨b) = ¬a ∧¬b ¬(a ∨b) = ¬a ∧¬b ¬(a ∧b) = ¬a ∨¬b ¬¬(a ∧b) = ¬¬a ∧¬¬b ¬¬a = a ¬¬¬a = ¬a ¬⊥ = ⊤,¬⊤ = ⊥ ¬⊥ = ⊤, ¬⊤ = ⊥

Fact

A semi-De Morgan algebra A is a De Morgan algebra if and only if A satisfies the equation a ∨b = ¬(¬a ∧¬b) = ¬¬(a ∨b). ¬¬(a ∧b) = ¬¬a ∧¬¬b and ¬¬¬a = ¬a can not be transformed into structural rules immediately!

Stratergy

◮ from semi-De Morgan algebras to construct heterogeneous semi-De

Morgan algebras in which every axiom is analytic

Stratergy

◮ from semi-De Morgan algebras to construct heterogeneous semi-De

Morgan algebras in which every axiom is analytic

◮ from heterogeneous semi-De Morgan algebras to construct semi-De

Morgan algebras

From single type to multi-type

Multi-type enviroment

Lemma

Given an SM-algebra L = (L,∧,∨,⊤,⊥,¬), let K := {¬¬a ∈ L | a ∈ L}. Define h : L ։ K and e : K ֒→ L by the assignments a → ¬¬a and α → α, respectively. Then for all α ∈ K and a ∈ L, h(e(α)) = α

Multi-type enviroment

Definition

For any SM-algebra L = (L,∧,∨,⊤,⊥,¬), let the kernel of L be the algebra KL = (K,∩,∪,∼,1,0) defined as follows:

• K1. K := Range(h), where h : L ։ K is defined by letting h(a) = ¬¬a

for any a ∈ L;

• K2. α∪β := h(¬¬(e(α)∨e(β))) for all α,β ∈ K;
• K3. α∩β := h(e(α)∧e(β)) for all α,β ∈ K;
• K4. 1 := h(⊤);
• K5. 0 := h(⊥);
• K6. ∼α := h(¬e(α)).

Multi-type enviroment

Lemma

For any SM-algebra L,

• 1. the kernel KL is a DM-algebra.
• 2. h is a lattice-homomorphism from L onto K, and for all α,β ∈ K,

e(α)∧e(β) = e(α∩β) e(1) = ⊤ e(0) = ⊥.

Heterogenous algebra

Definition

A heterogeneous SDM-algebra (HSM-algebra) is a tuple (L,A,e,h) satisfying the following conditions: H1 L is a bounded distributive lattice;

Heterogenous algebra

Definition

A heterogeneous SDM-algebra (HSM-algebra) is a tuple (L,A,e,h) satisfying the following conditions: H1 L is a bounded distributive lattice; H2 A is a De Morgan lattice;

Heterogenous algebra

Definition

A heterogeneous SDM-algebra (HSM-algebra) is a tuple (L,A,e,h) satisfying the following conditions: H1 L is a bounded distributive lattice; H2 A is a De Morgan lattice; H3 e : A ֒→ L is an order embedding, which satisfies: for all α1,α2 ∈ A, e(α1)∧e(α2) = e(α1 ∩α2) and e(1) = ⊤ and e(0) = ⊥

Heterogenous algebra

Definition

A heterogeneous SDM-algebra (HSM-algebra) is a tuple (L,A,e,h) satisfying the following conditions: H1 L is a bounded distributive lattice; H2 A is a De Morgan lattice; H3 e : A ֒→ L is an order embedding, which satisfies: for all α1,α2 ∈ A, e(α1)∧e(α2) = e(α1 ∩α2) and e(1) = ⊤ and e(0) = ⊥ H4 h : L ։ A is a lattice homomorphism;

Heterogenous algebra

Definition

A heterogeneous SDM-algebra (HSM-algebra) is a tuple (L,A,e,h) satisfying the following conditions: H1 L is a bounded distributive lattice; H2 A is a De Morgan lattice; H3 e : A ֒→ L is an order embedding, which satisfies: for all α1,α2 ∈ A, e(α1)∧e(α2) = e(α1 ∩α2) and e(1) = ⊤ and e(0) = ⊥ H4 h : L ։ A is a lattice homomorphism; H5 h(e(α)) = α for every α ∈ A. L A ∼ h e

From multi-type to single type

Heterogenous algebra

Lemma

If (L,D,e,h) is an heterogeneous SM-algebra, then L can be endowed with a structure of SM-algebra defining ¬ : L → L by ¬a := e(∼h(a)) for every a ∈ L. Moreover, D K.

Heterogenous algebra

Lemma

If (L,D,e,h) is an heterogeneous SM-algebra, then L can be endowed with a structure of SM-algebra defining ¬ : L → L by ¬a := e(∼h(a)) for every a ∈ L. Moreover, D K.

Definition

For any SM-algebra A, we let A+ = (L,K,h,e), where: · L is the lattice reduct of A; · K is the kernel of A; · e : K ֒→ L is defined by e(α) = α for all α ∈ K; · h : L ։ K is defined by h(a) = ¬¬a for all a ∈ L; For any HSM-algebra H, we let H+ = (L, ¬) where: · L is the distributive lattice of H; · ¬ : L → L is defined by the assignment a → e(∼h(a)) for all a ∈ L.

Heterogenous representation theory

For any SM-algebra A and any HSM-algebra H: A (A+)+ and H (H+)+.

Algebraic semantics for multi-type display calculus

Canonical extension

Definition

A HSM-algebra is perfect if:

• 1. both L and A are perfect;
• 2. e is an order-embedding and is completely meet-preserving;
• 3. h is a complete homomorphism.

Corollary

If (L,D,e,h) is an HSM-algebra, then (Lδ,Dδ,eπ,hδ) is a perfect HSM-algebra.

Canonical extension

L Lδ A Aδ ⊢ ⊣⊢ h

∼δ

∼ e′ h′ hδ eπ e

Corollary

If (L,¬) is an SM-algebra, then Lδ can be endowed with the structure of SM-algebra by defining ¬δ : Lδ → Lδ by ¬δ := eπ ◦∼δ ◦hδ. Moreover, Kδ

L KLδ.

Multi-type proper display calculus

Hilbert style semi-De Morgan logic

◮ the language L

A ::= p | ⊥ | ⊤ | ¬A | A ∧A | A ∨A

◮ Axioms

(A1) ⊥ ⊢ A (A2) A ⊢ ⊤ (A3) ¬⊤ ⊢ ⊥ (A4) ⊤ ⊢ ¬⊥ (A5) A ⊢ A (A6) A ∧B ⊢ A (A7) A ∧B ⊢ B (A8) A ⊢ A ∨B (A9) B ⊢ A ∨B (A10) ¬A ⊢ ¬¬¬A (A11) ¬¬¬A ⊢ ¬A (A12) ¬A ∧¬B ⊢ ¬(A ∨B) (A13) ¬¬A ∧¬¬B ⊢ ¬¬(A ∧B) (A14) A ∧(B ∨C) ⊢ (A ∧B)∨(A ∧C)

◮ Rules

R1. If A ⊢ B and B ⊢ C, then A ⊢ C; R2. If A ⊢ B and A ⊢ C, then A ⊢ B ∧C; R3. If A ⊢ B and C ⊢ B, then A ∨C ⊢ B; R4. If A ⊢ B, then ¬B ⊢ ¬A.

Multi-type Display calculus

◮ Structural and operational language of D.DL:

DL          A ::= p | ⊤ | ⊥ | α | A ∧A | A ∨A X ::= ˆ ⊤ | ˇ ⊥ | ˇ Γ | X ˆ ∧X | X ˇ ∨X | X ˆ > X | X ˇ →X

Multi-type Display calculus

◮ Structural and operational language of D.DL:

DL          A ::= p | ⊤ | ⊥ | α | A ∧A | A ∨A X ::= ˆ ⊤ | ˇ ⊥ | ˇ Γ | X ˆ ∧X | X ˇ ∨X | X ˆ > X | X ˇ →X

◮ Structural and operational language of D.DM:

DM          α ::= ◦A | 1 | 0 | ∼α | α∩α | α∪α Γ ::= ˜

• X | ˆ

1 | ˇ 0 | ˜ ∼Γ | Γ ˆ ∩Γ | Γ ˇ ∪Γ | Γ ˆ >

¬Γ | Γ ˇ

→¬ Γ

Interpretation

◮ Interpretation of structural DL connectives as their operational

counterparts DL connectives

categorization

f g

structural

ˆ ⊤ ˆ ∧ ˆ > ˇ ⊥ ˇ ∨ ˇ →

• perational

⊤ ∧ (> ) ⊥ ∨ (→)

adjoint pairs

ˆ ∧ ⊣ ˇ → ˆ > ⊣ ˇ ∨

Interpretation

◮ Interpretation of structural DL connectives as their operational

counterparts DL connectives

categorization

f g

structural

ˆ ⊤ ˆ ∧ ˆ > ˇ ⊥ ˇ ∨ ˇ →

• perational

⊤ ∧ (> ) ⊥ ∨ (→)

adjoint pairs

ˆ ∧ ⊣ ˇ → ˆ > ⊣ ˇ ∨

◮ Interpretation of structural DM connectives as their operational

counterparts DM connectives

categorization

f g f-g

structural

ˆ 1 ˆ ∩ ˆ >

¬

ˇ ˇ ∪ ˇ →¬ ˜ ∼

• perational

1 ∩ (>

¬)

∪ (→¬) ∼

adjoint pairs

ˆ ∩ ⊣ ˇ →¬ ˆ >

¬ ⊣ ˇ

∪ ˜ ∼ ⊣ ˜ ∼

Interpretation

◮ Interpretation of structural heterogeneous (from DL to DM and vice

versa) connectives as their operational counterparts DL → DM DM → DL DM → DL DL → DM

categorization

f-g f-g g f

structural

˜

• ˜
• ˇ
• ˆ
• perational
• adjoint pairs

˜

• ⊣ ˜
• ˆ

⊣ ˇ

Display Postulates

◮ DL-type display structural rules

X ˆ ∧Y ⊢ Z

res

Y ⊢ X ˇ →Z X ⊢ Y ˇ ∨Z

res

Y ˆ > X ⊢ Z

Display Postulates

◮ DL-type display structural rules

X ˆ ∧Y ⊢ Z

res

Y ⊢ X ˇ →Z X ⊢ Y ˇ ∨Z

res

Y ˆ > X ⊢ Z

◮ De Morgan lattice type display structural rules

˜ ∼Γ ⊢ ∆

adj

˜ ∼∆ ⊢ Γ Γ ⊢ ˜ ∼∆

adj

∆ ⊢ ˜ ∼Γ Γ ˆ ∩∆ ⊢ Θ

res

∆ ⊢ Γ ˇ →¬ Θ Γ ⊢ ∆ ˇ ∪Θ

res

∆ ˆ >

¬Γ ⊢ Θ

DL-type structural rules

Id p ⊢ p

X ⊢ A A ⊢ Y

Cut

X ⊢ Y X ⊢ Y

ˆ ⊤

X ˆ ∧ ˆ ⊤ ⊢ Y X ⊢ Y

ˇ ⊥

X ⊢ Y ˇ ∨ ˇ ⊥ X ˆ ∧Y ⊢ Z

E

Y ˆ ∧X ⊢ Z X ⊢ Y ˇ ∨Z

E

X ⊢ Z ˇ ∨Y (X ˆ ∧Y) ˆ ∧Z ⊢ W

A

X ˆ ∧(Y ˆ ∧Z) ⊢ Z X ⊢ (Y ˇ ∨Z) ˇ ∨W

A

X ⊢ Y ˇ ∨(Z ˇ ∨W) X ⊢ Y

W

X ˆ ∧Z ⊢ Y X ⊢ Y

W

X ⊢ Y ˇ ∨Z X ˆ ∧X ⊢ Y

C

X ⊢ Y X ⊢ Y ˇ ∨Y

C

X ⊢ Y

DM-type structural rules

Γ ⊢ α α ⊢ ∆

Cut

Γ ⊢ ∆ Γ ⊢ ∆

ˆ 1

Γ ˆ ∩ ˆ 1 ⊢ ∆ Γ ⊢ ∆

ˇ

Γ ⊢ ∆ ˇ ∪ ˇ Γ ˆ ∩∆ ⊢ Θ

E

∆ ˆ ∩Γ ⊢ Θ Γ ⊢ ∆ ˇ ∪Θ

E

Γ ⊢ Θ ˇ ∪∆ (Γ ˆ ∩∆) ˆ ∩Θ ⊢ Λ

A

Γ ˆ ∩(∆ ˆ ∩Θ) ⊢ Λ Γ ⊢ (∆ ˇ ∪Θ) ˇ ∪Λ

A

Γ ⊢ ∆ ˇ ∪(Θ ˇ ∪Λ) Γ ⊢ ∆

W

Γ ˆ ∩Θ ⊢ ∆ Γ ⊢ ∆

W

Γ ⊢ ∆ ˇ ∪Θ Γ ˆ ∩Γ ⊢ ∆

C

Γ ⊢ ∆ Γ ⊢ ∆ ˇ ∪∆

C

Γ ⊢ ∆ Γ ⊢ ∆

cont

˜ ∼∆ ⊢ ˜ ∼Γ

DL-type operational rules

ˆ ⊤ ⊢ X

⊤ ⊤ ⊢ X ⊤

ˆ ⊤ ⊢ ⊤

⊥ ⊥ ⊢ ˇ

⊥ X ⊢ ˇ ⊥

⊥

X ⊢ ⊥ A ˆ ∧B ⊢ X

∧ A ∧B ⊢ X

X ⊢ A Y ⊢ B

∧

X ˆ ∧Y ⊢ A ∧B A ⊢ X B ⊢ Y

∨

A ∨B ⊢ X ˇ ∨Y X ⊢ A ˇ ∨B

∨

X ⊢ A ∨B

DM-type operational rules

ˆ 1 ⊢ Γ

1 1 ⊢ Γ 1

ˆ 1 ⊢ 1 0 ⊢ ˇ Γ ⊢ ˇ Γ ⊢ 0 α ˆ ∩β ⊢ Γ

∩ α∩β ⊢ Γ

Γ ⊢ α ∆ ⊢ β

∩

Γ ˆ ∩∆ ⊢ α∩β α ⊢ Γ β ⊢ ∆

∪

α∪β ⊢ Γ ˇ ∪∆ Γ ⊢ α ˇ ∪β

∪

Γ ⊢ α∪β ˜ ∼α ⊢ Γ

∼ ∼α ⊢ Γ

Γ ⊢ ˜ ∼α

∼

Γ ⊢ ∼α

Multi-type rules

◮ Multi-type display postulates

X ⊢ ˇ Γ

adj

ˆ X ⊢ Γ ˜

• X ⊢ Γ

adj

X ⊢ ˜

• Γ

Multi-type rules

◮ Multi-type display postulates

X ⊢ ˇ Γ

adj

ˆ X ⊢ Γ ˜

• X ⊢ Γ

adj

X ⊢ ˜

• Γ

◮ Multi-type structural rules

X ⊢ Y

˜

• ˜
• X ⊢ ˜
• Y

Γ ⊢ ˜

• ˇ

∆

˜

• ˇ
• Γ ⊢ ∆

X ⊢ ˇ ˆ 1

ˇ ˆ 1 X ⊢ ˆ

⊤ X ⊢ ˇ ˇ

ˇ ˇ

X ⊢ ˇ ⊥

Multi-type rules

◮ Multi-type display postulates

X ⊢ ˇ Γ

adj

ˆ X ⊢ Γ ˜

• X ⊢ Γ

adj

X ⊢ ˜

• Γ

◮ Multi-type structural rules

X ⊢ Y

˜

• ˜
• X ⊢ ˜
• Y

Γ ⊢ ˜

• ˇ

∆

˜

• ˇ
• Γ ⊢ ∆

X ⊢ ˇ ˆ 1

ˇ ˆ 1 X ⊢ ˆ

⊤ X ⊢ ˇ ˇ

ˇ ˇ

X ⊢ ˇ ⊥

◮ Multi-type operational rules

˜

• A ⊢ Y
• A ⊢ Y

X ⊢ ˜

• A
• X ⊢ ◦A

A ⊢ X

A ⊢ ˇ

Y X ⊢ ˇ A

• X ⊢ A

Translation functions

The translations τ : L → LMT is defined by simultaneous induction as follows: pτ ::= p ⊤τ ::= ⊤ ⊥τ ::= ⊥ (A ∧B)τ ::= Aτ ∧Bτ (A ∨B)τ ::= Aτ ∨Bτ (¬A)τ ::= ∼◦Aτ

Example

¬¬A ∧¬¬B ⊢ ¬¬(A ∧B)

• ∼◦∼◦A ∧∼◦∼◦B ⊢ ∼◦∼◦(A ∧B)

Example

¬¬A ∧¬¬B ⊢ ¬¬(A ∧B)

• ∼◦∼◦A ∧∼◦∼◦B ⊢ ∼◦∼◦(A ∧B)

◮ Step 1: A ⊢ A ˜

• ˜
• A ⊢ ˜
• A
• A ⊢ ˜
• A

cont ˜ ∼˜

• A ⊢ ˜

∼◦A ˜ ∼˜

• A ⊢ ∼◦A

˜

• ˇ
• ˜

∼˜

• A ⊢ ˜
• ˇ

∼◦A ˜

• ˜

∼˜

• A ⊢ ˇ

∼◦A ˜

• ˜

∼˜

• A ⊢ ∼◦A

˜ ∼˜

• A ⊢ ˜
• ∼◦A

˜ ∼˜

• A ⊢ ◦∼◦A

˜ ∼◦∼◦A ⊢ ˜

• A

∼◦∼◦A ⊢ ˜

• A

∼◦∼◦A ⊢ ˇ ˜

• A

W ∼◦∼◦A ˆ ∧∼◦∼◦B ⊢ ˇ ˜

• A

∼◦∼◦A ∧∼◦∼◦B ⊢ ˇ ˜

• A

ˆ (∼◦∼◦A ∧∼◦∼◦B) ⊢ ˜

• A

˜

• ˆ

(∼◦∼◦A ∧∼◦∼◦B) ⊢ A

Example

¬¬A ∧¬¬B ⊢ ¬¬(A ∧B)

• ∼◦∼◦A ∧∼◦∼◦B ⊢ ∼◦∼◦(A ∧B)

◮ Step 1: A ⊢ A ˜

• ˜
• A ⊢ ˜
• A
• A ⊢ ˜
• A

cont ˜ ∼˜

• A ⊢ ˜

∼◦A ˜ ∼˜

• A ⊢ ∼◦A

˜

• ˇ
• ˜

∼˜

• A ⊢ ˜
• ˇ

∼◦A ˜

• ˜

∼˜

• A ⊢ ˇ

∼◦A ˜

• ˜

∼˜

• A ⊢ ∼◦A

˜ ∼˜

• A ⊢ ˜
• ∼◦A

˜ ∼˜

• A ⊢ ◦∼◦A

˜ ∼◦∼◦A ⊢ ˜

• A

∼◦∼◦A ⊢ ˜

• A

∼◦∼◦A ⊢ ˇ ˜

• A

W ∼◦∼◦A ˆ ∧∼◦∼◦B ⊢ ˇ ˜

• A

∼◦∼◦A ∧∼◦∼◦B ⊢ ˇ ˜

• A

ˆ (∼◦∼◦A ∧∼◦∼◦B) ⊢ ˜

• A

˜

• ˆ

(∼◦∼◦A ∧∼◦∼◦B) ⊢ A ◮ Step 2: ˜

• ˆ

(∼◦∼◦A ∧∼◦∼◦B) ⊢ B

Example

◮ Step 3: ˜

• ˆ

(∼◦∼◦A ∧∼◦∼◦B) ⊢ A ˜

• ˆ

(∼◦∼◦A ∧∼◦∼◦B) ⊢ B ˜

• ˆ

(∼◦∼◦A ∧∼◦∼◦B) ˆ ∧ ˜

• ˆ

(∼◦∼◦A ∧∼◦∼◦B) ⊢ A ∧B C ˜

• ˆ

(∼◦∼◦A ∧∼◦∼◦B) ⊢ A ∧B ˆ (∼◦∼◦A ∧∼◦∼◦B) ⊢ ˜

• (A ∧B)

ˆ (∼◦∼◦A ∧∼◦∼◦B) ⊢ ◦(A ∧B) cont ˜ ∼◦(A ∧B) ⊢ ˜ ∼ ˆ (∼◦∼◦A ∧∼◦∼◦B) ∼◦(A ∧B) ⊢ ˜ ∼ ˆ (∼◦∼◦A ∧∼◦∼◦B) ∼◦(A ∧B) ⊢ ˇ ˜ ∼ ˆ (∼◦∼◦A ∧∼◦∼◦B) ˜

• ˜
• ∼◦(A ∧B) ⊢ ˜
• ˇ

˜ ∼ ˆ (∼◦∼◦A ∧∼◦∼◦B) ˜

• ˇ
• ˜
• ∼◦(A ∧B) ⊢ ˜

∼ ˆ (∼◦∼◦A ∧∼◦∼◦B)

• ∼◦(A ∧B) ⊢ ˜

∼ ˆ (∼◦∼◦A ∧∼◦∼◦B) ˆ (∼◦∼◦A ∧∼◦∼◦B) ⊢ ˜ ∼◦∼◦(A ∧B) ˆ (∼◦∼◦A ∧∼◦∼◦B) ⊢ ∼◦∼◦(A ∧B) ∼◦∼◦A ∧∼◦∼◦B ⊢ ˇ ∼◦∼◦(A ∧B) ∼◦∼◦A ∧∼◦∼◦B ⊢ ∼◦∼◦(A ∧B)

Equality

Theorem

For all L-formulas A and B and every SM-algebra L, L |= A ≤ B iff L+ |= Aτ ≤ Bτ.

Properties

Theorem (Completeness)

D.SDM is complete with respect to the class of semi-De Morgan algebras.

Theorem (Conservative extension)

D.SDM is a conservative extension of H.SDM.

Theorem (Cut elimination)

If X ⊢ Y is derivable in D.SDM, then it is derivable without (Cut).

Theorem (Subformula property)

Any cut-free proof of the sequent X ⊢ Y in D.SDM contains only structures over subformulas of formulas in X and Y.

Future work

◮ extensions to other algebras based on semi-De Morgan algebras,

e.g. quasi-De Morgan algebras, demi-p-algebras, weak stone algebras, etc.;

Future work

◮ extensions to other algebras based on semi-De Morgan algebras,

e.g. quasi-De Morgan algebras, demi-p-algebras, weak stone algebras, etc.;

◮ compatebility frames for semi-De Morgan algebras

Thanks for your attention!